Real Valued Functions

Q.

If $f\left(x\right)=x^2$, find $\frac{f\left(1.1\right)-f\left(1\right)}{(1.1-1)}.$

Q.

The minimum and maximum values of $f\left(x\right)=x^2+4x+17$ are

1. 25 and 82

2. 13 and

3. 17 and 82

4. 15 and

Q. The number of values of $x$ satisfying the equation $1+x=\text{sgn}\left(x\right)$ is
(where $\text{sgn}\left(x\right)$ denotes the signum function)

Q.

Solution set of $\frac{(x-1)(x-2{)}^2(x-4)}{(x+2)(x-3)}\ge 0$ is :

1. $(-\infty ,-4)\cup \left\{2\right\}\cup (3,\infty )$

2. $(-4,2]\cup [1,2]\cup (3,\infty )$

3. $(-\infty ,-4)\cup \left\{2\right\}\cup (3,\infty )$

4. $Noneofthese$

Q. If $x^2-3x\text{sgn}\left(x\right)+2=0,$ then the value(s) of $x$ is/are
(where $\text{sgn}\left(x\right)$ denotes the signum function)

1. $-1$
2. $-2$
3. $1$
4. $2$

Q.

A hyperbola is given as $x^2.se{c}^2\alpha -y^2.cose{c}^2\alpha =1$

What's the eccentricty of the hyperbola?

1. 

2. 

3. 

4. 

Q. Let $f\left(x\right)$ be a real valued function. Then the domain of $f\left(x\right)=\sqrt{x-\sqrt{1-x^2}}$ is

1. $[-1,-\frac{1}{\sqrt{2}}]\cup [\frac{1}{\sqrt{2}},1]$
2. $(-\infty ,-\frac{1}{\sqrt{2}}]\cup [\frac{1}{\sqrt{2}},\infty )$
3. $[-1,1]$
4. $[\frac{1}{\sqrt{2}},1]$

Q. Which of the following functions are the mirror images of each other with respect to the line $y=x$

1. $e^x$ and ${\log }_{e}x$
2. $e^x$ and ${\log }_{x}e$
3. ${\pi }^x$ and ${\log }_{\pi }x$
4. ${\pi }^x$ and ${\log }_x\pi$

Q. Let $f\left(x\right)$ be a real valued function, then the number of integral values of $x$ for which $f\left(x\right)=\sqrt{x+2}+\sqrt{7-x}$ is defined, is

Q. $\mathrm{If}f\left(x\right)=\cos \left(\log x\right),x>0;\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}f\left(x\right)f\left(4\right)-\frac{1}{2}\left(f\left(\frac{x}{4}\right)+f\left(4x\right)\right\}=$

1. 1
2. -1
3. \N
4. $\pm 1$

Q.

The minimum value of f(x) = -2 $x^2$ + 5x + 4 ∀ x $\in$ [0, 3] is __

Q. If $f\left(x\right)=x^2-5x+6$ is a real-valued function, then

1. $f\left(x\right)\ge 0$ when $x\in (-\infty ,2]\cup [3,\infty )$
2. $f\left(x\right)\ge 0$ when $x\in [2,\infty )$
3. $f\left(x\right)<0$ when $x\in (-\infty ,2)$
4. $f\left(x\right)<0$ when $x\in (2,3)$

Q. If the domain of the function $f$ is $R$, then which of the following will not be a real valued function ?

1. $f\left(x\right)=x^2$
2. $f\left(x\right)=|x-2|$
3. $f\left(x\right)=5x-7$
4. $f\left(x\right)=\sqrt{x}$

Q. Consider the real-valued function $f$ satisfying $2f\left(\sin x\right)+f\left(\cos x\right)=x$. Then,

1. Domain of $f$ is $R$
2. Domain of $f$ is $[-1,1]$
3. Range of $f$ is $[-\frac{2\pi }{3},\frac{\pi }{3}]$
4. Range of $f$ is $R$

Q.

Find the sum to $n$ terms of the sequence ${\left(x+\frac{1}{x}\right)}^2,{\left(x^2+\frac{1}{x^2}\right)}^2,{\left(x^3+\frac{1}{x^3}\right)}^2,\dots$.

Q. Which of the following options is/are true for the function $f\left(x\right)=|x-3|+|x-4|$

1. The minimum value of $f\left(x\right)$ is $1$
2. $f\left(x\right)=2x-7$ when $x\ge 4$
3. $f\left(x\right)=1$ when $x<3$
4. The graph of $f\left(x\right)$ is
   

Q. If $\omega$ be a complex cube root of unity, then $\left|\begin{array}{ccc}1& \omega & -\frac{{\omega }^2}{2}\\ 1& 1& 1\\ 1& -1& 0\end{array}\right|=$

1. $\omega$
2. ${\omega }^2$
3. 0
4. 1

Q. Which of the following is (are) CORRECT for a real function?

1. Domain of a real function should be subset of $R$
2. Range of a real function should be subset of $R$
3. If $f:R\to R$ and $f\left(x\right)=1-x^2$, then it is real function
4. If $f:R\to R$ and $f\left(x\right)=x^3-2|x|$, then it is real function